Two centuries of trend following

[Pages:17]arXiv:1404.3274v1 [q-fin.PM] 12 Apr 2014

Two centuries of trend following

Y. Lemp?ri?re, C. Deremble, P. Seager, M. Potters, J. P. Bouchaud Capital Fund Management, 23 rue de l'Universit?, 75007 Paris, France

April 15, 2014

Abstract

We establish the existence of anomalous excess returns based on trend following strategies across four asset classes (commodities, currencies, stock indices, bonds) and over very long time scales. We use for our studies both futures time series, that exist since 1960, and spot time series that allow us to go back to 1800 on commodities and indices. The overall t-stat of the excess returns is 5 since 1960 and 10 since 1800, after accounting for the overall upward drift of these markets. The effect is very stable, both across time and asset classes. It makes the existence of trends one of the most statistically significant anomalies in financial markets. When analyzing the trend following signal further, we find a clear saturation effect for large signals, suggesting that fundamentalist traders do not attempt to resist "weak trends", but step in when their own signal becomes strong enough. Finally, we study the performance of trend following in the recent period. We find no sign of a statistical degradation of long trends, whereas shorter trends have significantly withered.

1 Introduction

Are markets efficient, in the sense that all public information is included in current prices? If this were so, price changes would be totally unpredictable in the sense that no systematic excess return based on public information can be achievable. After decades of euphoria in economics departments (There is no other proposition in economics which has more solid empirical evidence supporting it than the Efficient Market Hypothesis, as M. Jensen famously wrote in 1978), serious doubts were raised by behavioral economists who established a long series of pricing "anomalies" [1]. The most famous of these anomalies (and arguably the most difficult to sweep under the rug) is the so-called "excess volatility puzzle", unveiled by R. Shiller and others [2, 3]. Strangely (or wisely?) the 2013 Nobel committee decided not to take sides, and declared that markets are indeed efficient (as claimed by laureate E. Fama), but that the theory actually makes "little sense", as argued by Shiller, who shared the same prize!1. (See also [4, 5, 6] for insightful papers on this debate.)

In the list of long-known anomalies, the existence of trends plays a special role. First, because trending is the exact opposite of the mechanisms that should ensure that markets are efficient, i.e. reversion forces that drive prices back to the purported fundamental value. Second, because persistent returns validate

1Together with a third scientist, L. Hansen, who had not directly taken part in the debate

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a dramatically simple strategy, trend following, which amounts to buying when the price goes up, and selling when it goes down. Simple as it may be [7], this strategy is at the heart of the activity of CTAs (Commodity Trading Advisors [8]), an industry that now manages (as of Q4, 2013) no less than 325 B$, representing around 16% of the total amount of assets of the hedge fund industry, and accounting for several percent of the daily activity of futures markets.2 [9] These numbers are by no means small, and make it hard for efficient market enthusiasts to dismiss this anomaly as economically irrelevant.3 The strategy is furthermore deployed over a wide range of instruments (indices, bonds, commodities, currencies...) with positive reported performance over long periods, suggesting that the anomaly is to a large extent universal, both across epochs and asset classes.4 This reveals an extremely persistent, universal bias in the behaviour of investors who appear to hold "extrapolative expectations", as argued in many papers coming from different strands of the academic literature: see e.g. [12, 13, 14, 15, 16, 17, 18, 19, 20] and refs. therein.

Many academic studies have already investigated this trend anomaly on a wide range of assets, and have convincingly established its statistical significance in the last few decades [21, 22]. Recently, this time horizon has been extended to 100 years in ref. [23], and the effect still exists unabated. The aim of the present paper is to extend the time horizon even further, to 200 years, as far in the past as we have been able to go in terms of data. We find that the amplitude of the effect has been remarkably steady over two centuries. This also allows us to assess the recent weakening of the effect (as testified by the relatively poor performance of CTAs over the last five years). We show that the very recent past is fully compatible with a statistical fluctuation. Although we cannot exclude that this recent period is a precursor of the "end of trends", we argue theoretically that this is an unlikely scenario. We give several mechanisms that could explain the existence and persistence of these trends throughout history.

Note that trends not only exist for market factors such as indices, bonds, currencies, etc., but also cross-sectionally in stock markets. The so-called "momentum anomaly" consists in buying the past winners and selling the past losers in a market neutral way, with again a high statistical significance across many decades and different geographical zones [24, 25, 26], and [10] for a recent review. Although interesting in its own right (and vindicating the hypothesis that trend following is universal [27]), we will not study this particular aspect of trend following in the present paper.

The outline of the paper is as follows. In the next section, we define the trend following indicator used for this study, and test its statistical significance on available futures data. We start with futures since they are the preferred instruments of trend followers in finance. Also, their prices are unambiguously defined by transparent market trades, and not the result of a proprietary computation. In the following part, we carefully examine, for each asset class, how the available time-series can be extended as far in the past as possible. We then

2Futures markets allow one to go short as easily as to go long. Therefore both up-trends and down-trends can equally be exploited.

3M. Jensen (1978) actually stressed the importance of trading profitability in assessing market efficiency. In particular, if anomalous return behavior is not definitive enough for an efficient trader to make money trading on it, then it is not economically significant.

4Note that the excess return of trends cannot be classified as risk-premium either, see [10, 11]. On the contrary, trend following is correlated with "long-vol" strategies.

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present our results over two centuries, and show how exceptionally stable long trends have been. We examine more in depth the linearity of the signal, and find that the trend predictability in fact saturates for large values of the signal, which is needed for the long term stability of markets. We finally discuss the significance of the recent performance of the trend in light of this long-term simulation.

2 Trend-following on futures since 1960

2.1 Measuring trends

We choose to define our trend indicator in a way similar to simulating a constant risk trading strategy (without costs). More precisely, we first define the reference price level at time t, p n,t, as an exponential moving average of past prices (excluding p(t) itself) with a decay rate equal to n months. Long simulations can often only be performed on monthly data, so we use monthly closes. The signal sn(t) at the beginning of month t is constructed as:

sn(t)

=

p(t

- 1) - n(t -

p n,t-1 1)

,

(1)

where the volatility n is equal to the exponential moving average of the absolute monthly price changes, with a decay rate equal to n months. The average strength of the trend is then measured as the statistical significance of fictitious profits and losses (P&L) of a risk managed strategy that buys or sells (depending on the sign of sn) a quantity ?n-1 of the underlying contract :5

Qn (t)

=

t 2000

SR (T) 0.66 1.15 1.05 1.12 0.75

t-stat (T) 2.1 3.64 3.3 3.5 2.8

t-stat (T) 1.8 2.5 2.85 3.03 1.9

SR (?) 0.17 0.78 -0.03 0.79 0.68

t-stat (?) 0.5 2.5 -0.1 2.5 2.15

Table 3: Sharpe ratio and t-stat of the trend (T) for n = 5, of the de-biased trend (T) and of the drift component ? for each decade.

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indices and commodities, and government rates for bonds. We shall examine each sector independently. Before doing so, however, we should mention other important restrictions on the use of the historical data. First, we expect trends to develop only on freely-traded instruments, where price evolution is not distorted by state interventions. Also, we require a certain amount of liquidity, in order to have meaningful prices. These two conditions, free-floating and liquid assets, will actually limit us when we look back in the distant past.

3.1 Currencies

The futures time series goes back to 1973. In the previous period (1944-1971), the monetary system operated under the rules set out in the Bretton Woods agreements. According to these international treaties, the exchange rates were pegged to the USD (within a 1% margin), which remained the only currency that was convertible into gold at a fixed rate. Therefore, no trend can be expected on these time series where prices are limited to a small band around a reference value.

Prior to this, the dominant system was the Gold Standard. In this regime, the international value of a currency was determined by a fixed relationship with gold. Gold in turn was used to settle international accounts. In this regime as well, we cannot expect trends to develop, since the value of the currency is essentially fixed by its conversion rate with gold. In the 1930s, many countries dropped out of this system, massively devaluing in a desperate attempt to manage the consequences of the Great Depression (the "beggar thy neighbor" policy). This also led to massively managed currencies, with little hope of finding any genuine trending behavior.

All in all, therefore, it seems unlikely that one can find a free-floating substitute for our futures time-series on foreign exchange, prior to 1973.

3.2 Government rates

Government debt (and default!) has been around for centuries [28], but in order to observe a trend on interest rates, one needs a liquid secondary market, on which the debt can be exchanged at all times. This is a highly non-trivial feature for this market. Indeed, throughout most of the available history, government debt has been used mostly as a way to finance extraordinary liabilities, such as wars. In other periods of history, debt levels gradually reduced, as the principals were repaid, or washed away by growth (as debt levels are quoted relative to GDP).

As a typical example, one can see in Fig. 2 that the US debt, inherited from the War of Independence, practically fell to 0 in 1835-36, during the Jackson presidency. There is another spike in 1860-65, during the American Civil War, which then gets gradually washed away by growth. We have to wait until the first World War to see a significant increase in debt which then persists until today. Apart from Australia, whose debt has grown at a roughly constant rate, and Japan, whose turning point is around 1905 and the war with Russia, the situation in all other countries is similar to that of the US. From this point onwards, the debt has never been repaid in its totality in any of the countries we consider in this study, and has mostly been rolled from one bond issuance to the next.

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10000

100

1

0.01

0.0001

1800

1850

1900

1950

2000

1.2

1

0.8

0.6

0.4

0.2

0

1800

1850

1900

1950

2000

Figure 2: Global debt of the US government, in USD billions and as a fraction of GDP

country USA

Australia Canada Germany Switzerland Japan United Kingdom

start 1913 1911 1935 1914 1907 1904 1844

Table 4: Starting date of the central bank's monopoly on the issuance of notes. The bank of England does not have this monopoly in Scotland and Ireland, but regulates the commercial banks that share this privilege.

Another more subtle point can explain the emergence of a stable debt market: at the beginning of the XXth century, the monetary policy (in its most straightforward sense: the power to print money) was separated from the executive instances and attributed to central banks, supposedly independent from the political power (see Fig. 4). This move increased the confidence in the national debt of these countries, and helped boost subsequent debt levels.

All of this leads us to the conclusion that the bond market before 1918 was not developed enough to be considered as "freely traded and liquid". Therefore, we start our interest rate time-series in 1918. We should note as well that we exclude from the time-series the war and immediate post-war period in Japan and Germany, where the economy was heavily managed, therefore leading to price distortions.

3.3 Indices and commodities

For these sectors, the situation is more straightforward. Stocks and commodities have been actively priced throughout the XIXth century, so it is relatively easy to get clean, well-defined prices. As we can see from Tables 5 and 6, we can characterize trend following strategies for over more than 2 centuries on some of these time-series. Apart from some episodes that we excluded, like WWII in Germany or Japan where the stock market was closed, or the period through

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country USA

Australia Canada Germany Switzerland Japan United Kingdom

start 1791 1875 1914 1870 1914 1914 1693

Table 5: Starting date of the spot index monthly time-series for each country.

commodity Crude oil Natural Gas

Corn Wheat Sugar Live Cattle Copper

start 1859 1986 1858 1841 1784 1858 1800

Table 6: Starting date of the spot price for each commodity.

which the price of Crude was fixed (in the second half of the XXth century), the time-series are of reasonably good quality, i.e. prices are actually moving (no gaps) and there are no major outliers.

3.4 Validating the proxies

We now want to check that the time series selected above, essentially based on spot data on 10 year government bonds, indices and commodities, yield results that are very similar to the ones we obtained with futures. This will validate our proxies and allow us to extend, in the following section, our simulations to the pre-1960 period.

In Fig. 3 we show a comparison of the trend applied to futures prices and to spot prices in the period of overlapping coverage between the two data sets. From 1982 onwards we have futures in all 4 sectors and the correlation is measured to be 91%, which we consider to be acceptably high. We show the correlations per sector calculated since 1960 in Table 7 and observe that the correlation remains high for indices and bonds but is lower for commodities with a correlation of 65%. We know that the difference between the spot and futures prices is the so called "cost of carry" which is absent for the spot data, this additional term being especially significant and volatile for commodities. We find however that the level of correlation is sufficiently high to render the results meaningful. In any case the addition of the cost of carry can only improve the performance of the trend on futures and any conclusion regarding trends on spot data will be further confirmed by the use of futures data.

We therefore feel justified in using the spot data to build statistics over a long history. We believe that the performance will be close to (and in any case,

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